Please submit a html output of your R notebook and include summary statistics and explanation of the variables in the dataset. Also, please include an explanation of your model results.
library(data.table)
library(leaps)Problem 1
Use the College dataset from ISLR2 library and use best subset selection, forward and backward selection methods to predict the number of applications received using the other variables.
dt <- data.table(ISLR2::College)
head(dt)The ISLR2::College dataset contains “Statistics for a large number of US Colleges from the 1995 issue of US News and World Report.” If you want to learn more, I suggest visiting https://rdocumentation.org/packages/ISLR2/versions/1.3-1/topics/College.
Summary Stats
summary(dt) Private Apps Accept Enroll Top10perc Top25perc F.Undergrad
No :212 Min. : 81 Min. : 72 Min. : 35 Min. : 1.00 Min. : 9.0 Min. : 139
Yes:565 1st Qu.: 776 1st Qu.: 604 1st Qu.: 242 1st Qu.:15.00 1st Qu.: 41.0 1st Qu.: 992
Median : 1558 Median : 1110 Median : 434 Median :23.00 Median : 54.0 Median : 1707
Mean : 3002 Mean : 2019 Mean : 780 Mean :27.56 Mean : 55.8 Mean : 3700
3rd Qu.: 3624 3rd Qu.: 2424 3rd Qu.: 902 3rd Qu.:35.00 3rd Qu.: 69.0 3rd Qu.: 4005
Max. :48094 Max. :26330 Max. :6392 Max. :96.00 Max. :100.0 Max. :31643
P.Undergrad Outstate Room.Board Books Personal PhD Terminal
Min. : 1.0 Min. : 2340 Min. :1780 Min. : 96.0 Min. : 250 Min. : 8.00 Min. : 24.0
1st Qu.: 95.0 1st Qu.: 7320 1st Qu.:3597 1st Qu.: 470.0 1st Qu.: 850 1st Qu.: 62.00 1st Qu.: 71.0
Median : 353.0 Median : 9990 Median :4200 Median : 500.0 Median :1200 Median : 75.00 Median : 82.0
Mean : 855.3 Mean :10441 Mean :4358 Mean : 549.4 Mean :1341 Mean : 72.66 Mean : 79.7
3rd Qu.: 967.0 3rd Qu.:12925 3rd Qu.:5050 3rd Qu.: 600.0 3rd Qu.:1700 3rd Qu.: 85.00 3rd Qu.: 92.0
Max. :21836.0 Max. :21700 Max. :8124 Max. :2340.0 Max. :6800 Max. :103.00 Max. :100.0
S.F.Ratio perc.alumni Expend Grad.Rate
Min. : 2.50 Min. : 0.00 Min. : 3186 Min. : 10.00
1st Qu.:11.50 1st Qu.:13.00 1st Qu.: 6751 1st Qu.: 53.00
Median :13.60 Median :21.00 Median : 8377 Median : 65.00
Mean :14.09 Mean :22.74 Mean : 9660 Mean : 65.46
3rd Qu.:16.50 3rd Qu.:31.00 3rd Qu.:10830 3rd Qu.: 78.00
Max. :39.80 Max. :64.00 Max. :56233 Max. :118.00 df <- dewey::ifelsedata(data.frame(round(cor(dt[, !c("Private")]), 3)),
.85, "x >= y & x != 1", matchCols = FALSE)
rownames(df) <- colnames(df)
dfGGally::ggpairs(dt, mapping = ggplot2::aes(color = Private))
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These are the summary statistics. There’s nothing crazy about them in general. Maybe a few outliers. There aren’t a ton of things that are correlated at a rate at or higher than \(85\%\). It is all generally expected.
Normal
best_fit <- regsubsets(Apps ~ ., dt, nvmax = 17)
best_summary <- summary(best_fit)
data.table("BIC" = best_summary$bic,
"Cp" = best_summary$cp,
"r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]
par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")
par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")
normal <- as.formula("Apps ~ + Private + Accept + Enroll + Top10perc + Top25perc + Outstate + Room.Board + PhD + Expend + Grad.Rate")Forward
best_fit <- regsubsets(Apps ~ ., dt, nvmax = 17, method = "forward")
best_summary <- summary(best_fit)
data.table("BIC" = best_summary$bic,
"Cp" = best_summary$cp,
"r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]
par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")
par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")
forward <- as.formula("Apps ~ + Private + Accept + Enroll + Top10perc + Top25perc + Outstate + Room.Board + PhD + Expend + Grad.Rate")Backward
best_fit <- regsubsets(Apps ~ ., dt, nvmax = 17, method = "backward")
best_summary <- summary(best_fit)
data.table("BIC" = best_summary$bic,
"Cp" = best_summary$cp,
"r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]
par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")
par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")
backward <- as.formula("Apps ~ + Private + Accept + Enroll + Top10perc + Top25perc + Outstate + Room.Board + PhD + Expend + Grad.Rate")Conclusion
regs <- dewey::regsearch(dt, "Apps", colnames(dt[, !c("Apps")]), 1, 10, "gaussian", 0, FALSE, TRUE)[1] "Assembling regresions..."
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dewey <- as.formula("Apps ~ + Accept + Top10perc")# regsubsets produced the same arguments
forms <- c(normal, dewey)
lapply(forms, function(x) { summary(lm(formula = x, dt)) })[[1]]
Call:
lm(formula = x, data = dt)
Residuals:
Min 1Q Median 3Q Max
-5085.2 -439.2 -27.4 315.6 7848.6
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -100.51668 265.47592 -0.379 0.705069
PrivateYes -575.07061 132.52820 -4.339 1.62e-05 ***
Accept 1.58422 0.04011 39.500 < 2e-16 ***
Enroll -0.56221 0.11091 -5.069 5.02e-07 ***
Top10perc 49.13909 5.51638 8.908 < 2e-16 ***
Top25perc -13.86531 4.41751 -3.139 0.001762 **
Outstate -0.09466 0.01829 -5.176 2.89e-07 ***
Room.Board 0.16374 0.04668 3.508 0.000478 ***
PhD -10.01609 3.11921 -3.211 0.001378 **
Expend 0.07274 0.01142 6.370 3.26e-10 ***
Grad.Rate 7.33269 2.82114 2.599 0.009524 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1043 on 766 degrees of freedom
Multiple R-squared: 0.9283, Adjusted R-squared: 0.9274
F-statistic: 991.9 on 10 and 766 DF, p-value: < 2.2e-16
[[2]]
Call:
lm(formula = x, data = dt)
Residuals:
Min 1Q Median 3Q Max
-5334.2 -513.9 -16.7 325.1 9780.8
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -892.97561 77.89816 -11.46 <2e-16 ***
Accept 1.44004 0.01678 85.80 <2e-16 ***
Top10perc 35.83112 2.33210 15.36 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1125 on 774 degrees of freedom
Multiple R-squared: 0.9158, Adjusted R-squared: 0.9156
F-statistic: 4208 on 2 and 774 DF, p-value: < 2.2e-16The model from regsubsets is a little bit more accurate but comes at the cost of needing to include many more variables. If data collection was no issue, the regsubsets one is good, if data collection is an issue, then mine is much better. Basically, as the number of applications accepted increases by one, the number of applications received increases by \(1.44\). As the percent of new students that were in the top 10% of their high school class increases by one, the number of applications received increases by \(35.83\).
Problem 2
Use the Boston data to predict the per capita crime rate using best subset selection, forward and backward selection methods.
dt <- data.table(ISLR2::Boston)
head(dt)The ISLR2::Boston dataset contains “A data set containing housing values in 506 suburbs of Boston.” If you want to learn more, I suggest visiting https://rdocumentation.org/packages/ISLR2/versions/1.3-1/topics/Boston.
Summary Stats
summary(dt) crim zn indus chas nox rm age
Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000 Min. :0.3850 Min. :3.561 Min. : 2.90
1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02
Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000 Median :0.5380 Median :6.208 Median : 77.50
Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917 Mean :0.5547 Mean :6.285 Mean : 68.57
3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08
Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000 Max. :0.8710 Max. :8.780 Max. :100.00
dis rad tax ptratio lstat medv
Min. : 1.130 Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 1.73 Min. : 5.00
1st Qu.: 2.100 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.: 6.95 1st Qu.:17.02
Median : 3.207 Median : 5.000 Median :330.0 Median :19.05 Median :11.36 Median :21.20
Mean : 3.795 Mean : 9.549 Mean :408.2 Mean :18.46 Mean :12.65 Mean :22.53
3rd Qu.: 5.188 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:16.95 3rd Qu.:25.00
Max. :12.127 Max. :24.000 Max. :711.0 Max. :22.00 Max. :37.97 Max. :50.00 df <- dewey::ifelsedata(data.frame(round(cor(dt), 3)),
.85, "x >= y & x != 1", matchCols = FALSE)
rownames(df) <- colnames(df)
dfGGally::ggpairs(dt)
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plot: [13,11] [=========================================================================================>-] 99% est: 0s
plot: [13,12] [=========================================================================================>-] 99% est: 0s
plot: [13,13] [===========================================================================================]100% est: 0s
There’s nothing crazy with these numbers. It is weird that only tax and rad are correlated above \(85%\), but then again highways decrease property taxes or something. idk.
Normal
best_fit <- regsubsets(crim ~ ., dt, nvmax = 12)
best_summary <- summary(best_fit)
data.table("BIC" = best_summary$bic,
"Cp" = best_summary$cp,
"r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]
par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")
par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")
normal <- as.formula("crim ~ + zn + nox + dis + rad + ptratio + lstat + medv")Forward
best_fit <- regsubsets(crim ~ ., dt, nvmax = 12, method = "forward")
best_summary <- summary(best_fit)
data.table("BIC" = best_summary$bic,
"Cp" = best_summary$cp,
"r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]
par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")
par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")
forward <- as.formula("crim ~ + zn + nox + rm + dis + rad + ptratio + lstat + medv")Backward
best_fit <- regsubsets(crim ~ ., dt, nvmax = 12, method = "backward")
best_summary <- summary(best_fit)
data.table("BIC" = best_summary$bic,
"Cp" = best_summary$cp,
"r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]
par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")
par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")
backward <- as.formula("crim ~ zn + nox + dis + rad + ptratio + lstat + medv")Conclusion
regs <- dewey::regsearch(dt, "crim", c(colnames(dt[, !c("crim")]), "lstat*rad"), 1, 12, "gaussian", 0, FALSE, TRUE)[1] "Assembling regresions..."
| | 0 % ~calculating
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[1] "Creating 8190 formulas. Please be patient, this may take a while."
[1] "Creating regressions..."
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[1] "Running 5119 regressions. Please be patient, this may take a while."
[1] "Running regressions..."
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dewey <- as.formula("crim ~ + lstat + rad")# regsubsets produced the same arguments for normal and backward
# dropped backward
forms <- c(normal, forward, dewey)
lapply(forms, function(x) { summary(lm(formula = x, dt)) })[[1]]
Call:
lm(formula = x, data = dt)
Residuals:
Min 1Q Median 3Q Max
-8.655 -2.143 -0.319 1.050 74.740
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 17.46682 6.02424 2.899 0.003904 **
zn 0.04497 0.01803 2.494 0.012951 *
nox -12.45782 4.77637 -2.608 0.009375 **
dis -0.94255 0.26270 -3.588 0.000366 ***
rad 0.56152 0.04813 11.667 < 2e-16 ***
ptratio -0.34703 0.18288 -1.898 0.058322 .
lstat 0.11479 0.06945 1.653 0.098997 .
medv -0.19026 0.05369 -3.543 0.000432 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.452 on 498 degrees of freedom
Multiple R-squared: 0.4452, Adjusted R-squared: 0.4374
F-statistic: 57.08 on 7 and 498 DF, p-value: < 2.2e-16
[[2]]
Call:
lm(formula = x, data = dt)
Residuals:
Min 1Q Median 3Q Max
-8.724 -2.181 -0.288 1.081 73.947
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 13.25994 6.97229 1.902 0.057774 .
zn 0.04332 0.01807 2.397 0.016906 *
nox -12.50587 4.77446 -2.619 0.009080 **
rm 0.70899 0.59233 1.197 0.231896
dis -0.93005 0.26279 -3.539 0.000439 ***
rad 0.55378 0.04854 11.409 < 2e-16 ***
ptratio -0.33823 0.18294 -1.849 0.065079 .
lstat 0.13578 0.07160 1.896 0.058495 .
medv -0.21711 0.05817 -3.732 0.000212 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.449 on 497 degrees of freedom
Multiple R-squared: 0.4467, Adjusted R-squared: 0.4378
F-statistic: 50.17 on 8 and 497 DF, p-value: < 2.2e-16
[[3]]
Call:
lm(formula = x, data = dt)
Residuals:
Min 1Q Median 3Q Max
-8.953 -1.881 -0.249 1.040 76.726
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.38141 0.59872 -7.318 1.00e-12 ***
lstat 0.23728 0.04685 5.065 5.75e-07 ***
rad 0.52281 0.03842 13.607 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.559 on 503 degrees of freedom
Multiple R-squared: 0.4208, Adjusted R-squared: 0.4185
F-statistic: 182.7 on 2 and 503 DF, p-value: < 2.2e-16Again, regsubsets produces slightly better models, but mine is almost as good and is more parsimonious. As lstat increases by one, crim increases by \(.237\) and when rad increases by one, crim increases by \(.522\).
---
title: "Assignment 5"
author: "Gus Lipkin ~ glipkin6737@floridapoly.edu"
output:
  html_notebook:
    toc: true
    toc_depth: 2
    toc_float: true
  pdf_document: default
---

> Please submit a html output of your R notebook and include summary statistics and explanation of the variables in the dataset. Also, please include an explanation of your model results. 

```{r}
library(data.table)
library(leaps)
```

# Problem 1
> Use the College dataset from ISLR2 library and use best subset selection, forward and backward selection methods to predict the number of applications received using the other variables. 

```{r}
dt <- data.table(ISLR2::College)
head(dt)
```

The `ISLR2::College` dataset contains "Statistics for a large number of US Colleges from the 1995 issue of US News and World Report." If you want to learn more, I suggest visiting [https://rdocumentation.org/packages/ISLR2/versions/1.3-1/topics/College](https://rdocumentation.org/packages/ISLR2/versions/1.3-1/topics/College).

## Summary Stats
```{r warning=FALSE}
summary(dt)
df <- dewey::ifelsedata(data.frame(round(cor(dt[, !c("Private")]), 3)), 
                        .85, "x >= y & x != 1", matchCols = FALSE)
rownames(df) <- colnames(df)
df
GGally::ggpairs(dt, mapping = ggplot2::aes(color = Private))
```

These are the summary statistics. There's nothing crazy about them in general. Maybe a few outliers. There aren't a ton of things that are correlated at a rate at or higher than $85\%$. It is all generally expected.

## Normal
```{r}
best_fit <- regsubsets(Apps ~ ., dt, nvmax = 17)
best_summary <- summary(best_fit)

data.table("BIC" = best_summary$bic,
           "Cp" = best_summary$cp,
           "r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]

par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")

par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")

normal <- as.formula("Apps ~ + Private + Accept + Enroll + Top10perc + Top25perc + Outstate + Room.Board + PhD + Expend + Grad.Rate")
```

## Forward
```{r}
best_fit <- regsubsets(Apps ~ ., dt, nvmax = 17, method = "forward")
best_summary <- summary(best_fit)

data.table("BIC" = best_summary$bic,
           "Cp" = best_summary$cp,
           "r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]

par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")

par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")

forward <- as.formula("Apps ~ + Private + Accept + Enroll + Top10perc + Top25perc + Outstate + Room.Board + PhD + Expend + Grad.Rate")
```

## Backward
```{r}
best_fit <- regsubsets(Apps ~ ., dt, nvmax = 17, method = "backward")
best_summary <- summary(best_fit)

data.table("BIC" = best_summary$bic,
           "Cp" = best_summary$cp,
           "r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]

par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")

par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")

backward <- as.formula("Apps ~ + Private + Accept + Enroll + Top10perc + Top25perc + Outstate + Room.Board + PhD + Expend + Grad.Rate")
```

## Conclusion
```{r}
regs <- dewey::regsearch(dt, "Apps", colnames(dt[, !c("Apps")]), 1, 10, "gaussian", 0, FALSE, TRUE)
regs

dewey <- as.formula("Apps ~ + Accept + Top10perc")
```

```{r}
# regsubsets produced the same arguments
forms <- c(normal, dewey)

lapply(forms, function(x) { summary(lm(formula = x, dt)) })
```

The model from `regsubsets` is a little bit more accurate but comes at the cost of needing to include many more variables. If data collection was no issue, the `regsubsets` one is good, if data collection is an issue, then mine is much better. Basically, as the number of applications accepted increases by one, the number of applications received increases by $1.44$. As the percent of new students that were in the top 10% of their high school class increases by one, the number of applications received increases by $35.83$.

# Problem 2
> Use the Boston data to predict the per capita crime rate using  best subset selection, forward and backward selection methods.

```{r}
dt <- data.table(ISLR2::Boston)
head(dt)
```

The `ISLR2::Boston` dataset contains "A data set containing housing values in 506 suburbs of Boston." If you want to learn more, I suggest visiting [https://rdocumentation.org/packages/ISLR2/versions/1.3-1/topics/Boston](https://rdocumentation.org/packages/ISLR2/versions/1.3-1/topics/Boston).

## Summary Stats
```{r warning=FALSE}
summary(dt)
df <- dewey::ifelsedata(data.frame(round(cor(dt), 3)), 
                        .85, "x >= y & x != 1", matchCols = FALSE)
rownames(df) <- colnames(df)
df
GGally::ggpairs(dt)
```

There's nothing crazy with these numbers. It is weird that only `tax` and `rad` are correlated above $85%$, but then again highways decrease property taxes or something. idk.

## Normal
```{r}
best_fit <- regsubsets(crim ~ ., dt, nvmax = 12)
best_summary <- summary(best_fit)

data.table("BIC" = best_summary$bic,
           "Cp" = best_summary$cp,
           "r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]

par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")

par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")

normal <- as.formula("crim ~ + zn + nox + dis + rad + ptratio + lstat + medv")
```

## Forward
```{r}
best_fit <- regsubsets(crim ~ ., dt, nvmax = 12, method = "forward")
best_summary <- summary(best_fit)

data.table("BIC" = best_summary$bic,
           "Cp" = best_summary$cp,
           "r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]

par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")

par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")

forward <- as.formula("crim ~ + zn + nox + rm + dis + rad + ptratio + lstat + medv")
```

## Backward
```{r}
best_fit <- regsubsets(crim ~ ., dt, nvmax = 12, method = "backward")
best_summary <- summary(best_fit)

data.table("BIC" = best_summary$bic,
           "Cp" = best_summary$cp,
           "r2" = best_summary$adjr2)[order(r2 * -1, BIC, Cp)]

par(mfrow = c(1,2))
plot(best_summary$cp, xlab = "number of features", ylab = "cp")
plot(best_fit, scale = "Cp")

par(mfrow = c(1, 2))
plot(best_summary$bic, xlab = "number of features", ylab = "bic")
plot(best_fit, scale = "bic")

backward <- as.formula("crim ~ zn + nox + dis + rad + ptratio + lstat + medv")
```

## Conclusion
```{r}
regs <- dewey::regsearch(dt, "crim", c(colnames(dt[, !c("crim")]), "lstat*rad"), 1, 12, "gaussian", 0, FALSE, TRUE)
regs

dewey <- as.formula("crim ~ + lstat + rad")
```

```{r}
# regsubsets produced the same arguments for normal and backward
# dropped backward
forms <- c(normal, forward, dewey)

lapply(forms, function(x) { summary(lm(formula = x, dt)) })
```

Again, `regsubsets` produces slightly better models, but mine is almost as good and is more *parsimonious*. As `lstat` increases by one, `crim` increases by $.237$ and when `rad` increases by one, `crim` increases by $.522$.
